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Quantum ComputingMeghaditya Roy Chaudhury BCSE – IV Roll – 000810501052 Jadavpur University
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Overview Definition of Quantum Computing. Why Quantum Computing is necessary? Advantages over Classical Computation Quantum Algorithm: Shor’s Algorithm Current Developments and Future Prospects
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What is Quantum Computing? A quantum computer is a machine that performs calculations based on the laws of quantum mechanics, which is the behavior of particles at the sub-atomic level.
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Moore’s Law Moores law was a statement made in 1965 byGordon Moore, one of the founders of Intel. Moore noted that the number of transistorsthat could be squeezed on to a silicon chip wasdoubling every year. Over time, this has beenrevised to doubling every 18 months.This has held true …….. So far
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Problems At current rate transistors will be as small as an atom. If scale becomes too small, Electrons tunnel through micro-thin barriers between wires corrupting signals.
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Quantum Computing TimelineThe story of quantum computation started as early as1982, when the physicist Richard Feynmanconsidered simulation of quantum-mechanical objectsby other quantum systems1985 when David Deutsch of the University of Oxfordpublished a crucial theoretical paper in which hedescribed a universal quantum computer.In 1994 when Peter Shor from AT&Ts BellLaboratories in New Jersey devised the first quantumalgorithm.
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Nobody understands Quantum Mechanics“We always have had a great deal of difficultyin understanding the world view thatquantum mechanics represents ”- Richard Feynman ("Simulating physics with computers" ,1982)
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Representation of Data - Qubits A bit of data is represented by a single atom that is in one of two states denoted by |0> and |1>. A single bit of this form is known as a qubit A physical implementation of a qubit could use the two energy levels of an atom. An excited state representing |1> and a ground state representing |0>. Light pulse of frequency λ for Excited time interval t State NucleusGround State Electron State |0> State |1>
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Properties Of Quantum Mechanics Quantum Superposition Quantum Entanglement
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Representation of Data - SuperpositionA single qubit can be forced into a superposition of the two statesdenoted by the addition of the state vectors: ψ |ψ> = α 1 |0> + α 2 |1> αWhere α 1 and α 2 are complex numbers and |α 1| 2 + | α 2 | 2 = 1 A qubit in superposition is in both of the states |1> and |0> at the same time
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Relationships among data - Entanglement Entanglement is the ability of quantum systems to exhibitcorrelations between states within a superposition. Imagine two qubits, each in the state |0> + |1> (a superpositionof the 0 and 1.) We can entangle the two qubits such that themeasurement of one qubit is always correlated to themeasurement of the other qubit.
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Classical computation vs. Quantum ComputationClassical Computation Quantum Computation Data unit: bit Data unit: qubit = ‘1’ = ‘0’ =|1〉 =|0〉 Valid states: Valid states: x = ‘0’ or ‘1’ |ψ〉 = c1|0〉 + c2|1〉 x=0 x=1 |ψ〉 = |0〉 |ψ〉 = |1〉 |ψ〉 = (|0〉 + |1〉)/√2 0 0 1 1
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Classical computation vs. Quantum ComputationClassical Computation Quantum ComputationMeasurement: deterministic Measurement: stochasticState Result of measurement State Result of measurementx = ‘0’ ‘0’ |ψ〉 = |0〉 ‘0’x = ‘1’ ‘1’ |ψ〉 = |1〉 ‘1’ |ψ〉 = |0〉 + |1〉 ‘0’ 50% √2 ‘1’ 50%
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Quantum Algorithm:Shor’s Algorithm Shors algorithm is a quantum algorithm for factoring a number N in O((log N)3) time and O(log N) space, named after Peter Shor. The algorithm is significant because it implies that RSA, a popular public-key cryptography method, might be easily broken, given a sufficiently large quantum computer Like many quantum computer algorithms, Shors algorithm is probabilistic
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Quantum Algorithm:Shor’s Algorithm Shors algorithm consists of two parts: A reduction, which can be done on a classical computer, of the factoring problem to the problem of order-finding. f(x) = axmod(N) A quantum algorithm to solve the order-finding problem The algorithm is dependant on Modular Arithmetic Quantum Parallelism Quantum Fourier Transform
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Quantum Algorithm: Shor’s Algorithm In 2001, Shors algorithm was demonstrated by a group at IBM, who factored 15 into 3 × 5, using an NMR implementation of a quantum computer with 7 qubits with a classical computer# bits 1024 2048 4096factoring in 2006 105 years 5x1015 years 3x1029 yearsfactoring in 2024 38 years 1012 years 7x1025 yearsfactoring in 2042 3 days 3x108 years 2x1022 years with potential quantum computer# bits 1024 2048 4096# qubits 5124 10244 20484# gates 3x109 2X1011 X1012factoring time 4.5 min 36 min 4.8 hours
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Quantum computing incomputational complexity theory The class of problems that can be efficiently solved by quantum computers is called BQP, for "bounded error, quantum, polynomial time".
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Practical Implementations Ion Traps Nuclear magnetic resonance (NMR) Optical photon computer Solid-state